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3-202 A circuit board houses electronic components on one side, dissipating a total of 15 W through the backside of the
board to the surrounding medium. The temperatures on the two sides of the circuit board are to be determined for the cases of
no fins and 20 aluminum fins of rectangular profile on the backside.
Assumptions 1 Steady operating conditions exist. 2 The temperature in the board and along the fins varies in one direction
only (normal to the board). 3 All the heat generated in the chips is conducted across the circuit board, and is dissipated from
the backside of the board. 4 Heat transfer from the fin tips is negligible. 5 The heat transfer coefficient is constant and
uniform over the entire fin surface. 6 The thermal properties of the fins are constant. 7 The heat transfer coefficient accounts
for the effect of radiation from the fins.
Properties The thermal conductivities are given to be k = 12 W/m°C for the circuit board, k = 237 W/m°C for the aluminum
plate and fins, and k = 1.8 W/m°C for the epoxy adhesive.
Analysis (a) The thermal resistance of the board and the convection resistance
on the backside of the board are
T2
11
C/W 011.0
)m 15.0)(m 1.0(C) W/m.12(
m 002.0
board
=
==
kA
L
R
Rboard
T1
T
Rconv
3-162
3-203 An electronic device is cooled by dissipating heat through a heat sink attached on its top surface. There is contact
resistance at the interface of the electronic component and the heat sink. The surface temperature of the electronic device is to
be determined whether it is below 85°C or not.
Assumptions 1 Steady operating conditions exist. 2 Heat transfer is one-dimensional.3The electronic device maintains a
constant surface temperature.
PropertiesThe thermal contact conductance for aluminum plates with air at the interface, a roughness of about 10 μm and an
average interface pressure of 1 atm is hc = 3640 W/m2K (Table 3-1), the combined thermal resistance of an HS 5030 heat
sink, attached horizontally, is 1.2 K/W (Table 3-6).
AnalysisThe thermal resistances of different layers are
11
Since the surface temperature of the electronic device is above 85°C, there is a risk of overheating. To reduce the surface
temperature, the total thermal resistance needs to be reduced to promote more heat dissipation through the heat sink. One way
to solve this problem is by reducing the contact resistance at the interface. This can be achieved by filling the interface with a
fluid having higher thermal contact conductance than air.
sinkheat
total 1R
Ah
TT
R
T
c
s
+
−
1
−
−
−
Q
TT
K W/m250,11
W
K
2.1
45
3085
)m 040.0)(m 100.0( 2
1
=
−
−
=
−
c
h
Thus, the surface temperature of the electronic device can be reduced to below 85°C by filling the interface with a fluid
having a thermal contact conductance value higher than 11,250 W/m2K. From Table 3-1 hydrogen, silicone oil and glycerin
all have thermal contact conductance greater than 11,250 W/m2K.
DiscussionIn practice, the interfaces between electronic devices and heat sinks are filled with thermally conductive epoxy
adhesives to reduce thermal contact resistance.
3-163
3-204 Steam passes through a row of 10 parallel pipes placed horizontally in a concrete floor exposed to room air at 24
C
with a heat transfer coefficient of 12 W/m2.C. If the surface temperature of the concrete floor is not to exceed 38
C
, the
minimum burial depth of the steam pipes below the floor surface is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Heat transfer is two-dimensional (no change in the axial direction). 3
Thermal conductivity of the concrete is constant.
Properties The thermal conductivity of concrete is given to be k = 0.75 W/m°C.
Analysis In steady operation, the rate of heat loss from the steam through the
concrete floor by conduction must be equal to the rate of heat transfer from
the concrete floor to the room by combined convection and radiation, which
is determined to be
)(
−=
TThAQss
cm 22.2==⎯→⎯
=
m 222.0
)m 1(
2
sinh
m) (0.06
m) 2(1
ln
m) 5(2
m 47.10
z
z
10 m
Room
24C
38C
3-164
3-205 A cylindrical tank containing liquefied natural gas (LNG) is placed at the center of a square solid bar. The rate of heat
transfer to the tank and the LNG temperature at the end of a one–month period are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Heat transfer is two-dimensional (no change in the axial direction). 3
Thermal conductivity of the bar is constant. 4 The tank surface is at the same temperature as the LNG.
Properties The thermal conductivity of the bar is given to be k = 0.0002 W/m°C. The density and the specific heat of LNG
are given to be 425 kg/m3 and 3.475 kJ/kg°C, respectively,
Analysis The shape factor for this configuration is given in Table 3-7 to be
m 92.12
m 6.0
m 4.1
08.1ln
)m 9.1(2
08.1
ln
2=
=
=
D
w
L
S
C153.1−=
2
2
T
–160C
L = 1.9 m
D = 0.6 m
12C
1.4 m
3-165
3-166
3-207 A spherical tank containing iced water is buried underground. The rate of heat transfer to the tank is to be determined
for the insulated and uninsulated ground surface cases.
Assumptions 1 Steady operating conditions exist. 2 Heat transfer is two-dimensional (no change in the axial direction). 3
Thermal conductivity of the soil is constant. 4 The tank surface is assumed to be at the same temperature as the iced water
because of negligible resistance through the steel.
Properties The thermal conductivity of the soil is given to be k = 0.55 W/m°C.
Analysis The shape factor for this configuration is given in Table 3-7 to be
)m 2.2(2
2=
D
T1 =18C
3-167
Fundamentals of Engineering (FE) Exam Problems
3-209 Heat is lost at a rate of 275 W per m2 area of a 15-cm-thick wall with a thermal conductivity of k=1.1 W/mºC. The
temperature drop across the wall is
(a) 37.5ºC (b) 27.5ºC (c) 16.0ºC (d) 8.0ºC (e) 4.0ºC
3-210 Consider a wall that consists of two layers, A and B, with the following values: kA = 1.2 W/mºC, LA = 8 cm, kB = 0.2
W/mºC, LB = 5 cm. If the temperature drop across the wall is 18ºC, the rate of heat transfer through the wall per unit area of
the wall is
(a) 56.8 W/m2 (b) 72.1 W/m2 (c) 114 W/m2 (d) 201 W/m2 (e) 270 W/m2
3-168
3-211 Heat is generated steadily in a 3-cm-diameter spherical ball. The ball is exposed to ambient air at 26ºC with a heat
transfer coefficient of 7.5 W/m2ºC. The ball is to be covered with a material of thermal conductivity 0.15 W/mºC. The
thickness of the covering material that will maximize heat generation within the ball while maintaining ball surface
temperature constant is
(a) 0.5 cm (b) 1.0 cm (c) 1.5 cm (d) 2.0 cm (e) 2.5 cm
3-212 Consider a 1.5-m-high and 2-m-wide triple pane window. The thickness of each glass layer (k = 0.80 W/m.C) is 0.5
cm, and the thickness of each air space (k = 0.025 W/m.C ) is 1.2 cm. If the inner and outer surface temperatures of the
window are 10C and 0C, respectively, the rate of heat loss through the window is
(a) 3.4 W (b) 10.2 W (c) 30.7 W (d) 61.7 W (e) 86.8 W
3-169
3-213 Consider two metal plates pressed against each other. Other things being equal, which of the measures below will
cause the thermal contact resistance to increase?
(a) Cleaning the surfaces to make them shinier
(b) Pressing the plates against each other with a greater force
(c) Filling the gab with a conducting fluid
(d) Using softer metals
(e) Coating the contact surfaces with a thin layer of soft metal such as tin
3-214 A 10-m-long, 8-cm-outer-radius cylindrical steam pipe is covered with 3-cm thick cylindrical insulation with a thermal
conductivity of 0.05 W/m.C. If the rate of heat loss from the pipe is 1000 W, the temperature drop across the insulation is
(a) 58C (b) 101C (c) 143C (d) 282C (e) 600C
3-170
3-215 A 5-m diameter spherical tank is filled with liquid oxygen (ρ = 1141 kg/m3, cp = 1.71 kJ/kgºC) at -184ºC. It is
observed that the temperature of oxygen increases to -183ºC in a 144-hour period. The average rate of heat transfer to the
tank is
(a) 124 W (b) 185 W (c) 246 W (d) 348 W (e) 421 W
3-216 A 2.5-m-high, 4-m-wide, and 20-cm-thick wall of a house has a thermal resistance of 0.025ºC/W. The thermal
conductivity of the wall is
(a) 0.8 W/mºC (b) 1.2 W/mºC (c) 3.4 W/mºC (d) 5.2 W/mºC (e) 8.0 W/mºC
3-171